# Questions tagged [gauss-sums]

For questions on Gauss sums, a particular kind of finite sum of roots of unity.

82 questions
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### What is the Gauss sum equivalent of $\Gamma(s+1) = s\Gamma(s)$?

Gauss sums are analogous to the Gamma function: fix a complex number $s$ with real part $>0$. Then we have a multiplicative character $\chi_s :\mathbf R^{\times}_{>0} \to \mathbf C^\times$ given ...
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### How to prove that below quantity is purely imaginary?

How do I prove that the following quantity is purely imaginary: $$\sum_{0\leq l_1<l_2<l_3<l_4\leq q-1} e^{-2\pi i \frac{(l_1^2-l_2^2+l_3^2-l_4^2)}{q} }$$ where $q$ is an odd number?
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### Gauss-type sums for cube roots

(Quadratic) Gauss sums express square root of any integer as a sum of roots of unity (or of cosines of rational multiples of $2\pi$, if you will) with rational coefficients. But Kronecker-Weber ...
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### Prime power Gauss sums are zero

Fix an odd prime $p$. Then for a positive integer $a$, I can look at the quadratic Legendre symbol Gauss sum $$G_p(a) = \sum_{n \,\bmod\, p} \left( \frac{n}{p} \right) e^{2 \pi i a n / p}$$ where ...
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### Relation that holds for the Legendre symbol of an integer but not for the Jacobi symbol?

Let $p$ be a prime number and $\big(\frac{a}{p} \big)$ the Legendre symbol. Then we have the equality $$\sum_{a=1}^{p-1} \big(\frac{a}{p} \big) \zeta^a =\sum_{t=0}^{p-1} \zeta^{t^2},$$ where $\zeta$ ...
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### Gauss Sum of a Field with Four Elements

I need to calculate a couple of Gauss sums to solve a problem I'm working on, but I keep getting the wrong answer because the absolute value of what I calculate is impossible for such a sum. Can ...
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### Dirichlet character modulo p

How can I prove that if $\chi$ is a non-principal character modulo $p$ prime, then $\chi (-1) = \overline{\chi} (-1)= \pm 1$ and $\sum_{x=1}^p \chi (x) e^{2\pi i x}=0$? For the first question, I just ...
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### Finding closed form of a summation $\sum_{i = \lfloor{\frac{n}{2}}\rfloor}^{n}i$

So i'm trying to find the clsoed form of $\sum_{i = \lfloor{\frac{n}{2}}\rfloor}^{n}i$ from what I know its probably going to be $? * (\lfloor{\frac{n}{2}}\rfloor + n)$ based on when I wrote out ...
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### Can a bound be given to $\sum_{a=1}^{p-1}\chi(a)\left(\frac{a^2-1}{p}\right)$, which is smaller than $p$, where $\chi$ is a dirichlet character?

Here $(\cdot)$ denotes the legendre symbol and $p$ be an odd prime number. All terms are of norm 1. So one bound is $p$. can one better bound be given to the sum?
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It is well known the case for sums like: $$\sum_{i=0}^{p^n -1}\zeta^{-ai},$$ where zeta is a primitive $p^n$-rooth of $1$. But, is there a standard formula for sums like: $$\sum_{i=0}^{p^n -1}i^... 0answers 10 views ### Jacobi sums Gaussian Sum. Show J(\rho^{'},\chi^{'}) = (- J(\rho,\chi))^s I want to show that J(\rho^{'},\chi^{'}) = (- J(\rho,\chi))^s, where \rho^{'},\chi^{'} are characters of a finite field F_{p^s} and \chi,\rho are characters a finite field F_p. My work: I ... 1answer 46 views ### Gauss sum possible typo Let ψ: \mathbb{F_p} \to Z_p with the property ψ(a+b)=ψ(a)ψ(b) where Z_p denotes the p-adic integers. Assume further that ψ is not trivial. I'm trying to follow my professor's work, but I ... 0answers 60 views ### Upper bound on the summation of roots of unity Let p be a prime number, and \zeta = e^{(2\pi i/p)} be a pth root of unity. Given the gauss sum:$$g_t = \sum_{k = 0}^{p-1}\left(\frac{k}{p}\right)\zeta^{tk}$$I'm trying to prove the upper ... 0answers 25 views ### Gauss sum variant:  \sum_{ a, b \in \mathbb{Z} } e^{-i \pi (a^2+b^2)/p } \big( e^{\pi(a-b)} + e^{-\pi(a-b)} \big)  Motivated by an example in Chern Simons theory, let p \in \mathbb{Z} be prime, can anyone compute this sum:$$ \sum_{ a, b \in \mathbb{Z} } e^{-i \pi (a^2+b^2)/p } \big( e^{\pi(a-b)} + e^{-\pi(a-b)...
I have a $1\times k$ matrix representing $z$-scores and each element is correlated to each other according to a covariance matrix $\Sigma$. I would like to compute their sum of square and to know the ...
### Upper bound for $\frac{\|x\|_1}{\|x\|_2}$ if each entry of $x\in R^d$ is i.i.d. sampled from Gaussian distribution $N(0,1)$
In the question, $\|x\|_1=\sum_{i=1}^d|x_i|$ with $|\cdot|$ being the absolute value, and $\|x\|_2=\sqrt{\sum_{i=1}^d x_i^2}$. In general, $\frac{\|x\|_1}{\|x\|_2}\leq \sqrt{d}$ always holds for ...